On Tuza's conjecture for triangulations and graphs with small treewidth

Abstract

Tuza (1981) conjectured that the size $τ(G)$ of a minimum set of edges that intersects every triangle of a graph $G$ is at most twice the size $ν(G)$ of a maximum set of edge-disjoint triangles of $G$. In this paper we present three results regarding Tuza's Conjecture. We verify it for graphs with treewidth at most $6$; we show that $τ(G)\leq \frac{3}{2}\,ν(G)$ for every planar triangulation $G$ different from $K_4$; and that $τ(G)\leq\frac{9}{5}\,ν(G) + \frac{1}{5}$ if $G$ is a maximal graph with treewidth 3. Our first result strengthens a result of Tuza, implying that $τ(G) \leq 2\,ν(G)$ for every $K_8$-free chordal graph $G$.